Higher dimensional Sacks-Uhlenbeck-type functionals and applications

TL;DR

This paper extends the existence of harmonic spheres to higher dimensions using generalized Sacks-Uhlenbeck functionals, addressing regularity and bubbling issues through refined analysis and min-max methods.

Contribution

It introduces a new framework for constructing higher-dimensional harmonic spheres and develops advanced analytical tools to handle degeneracies and bubbling phenomena.

Findings

Constructed regular, non-trivial n-harmonic n-spheres for n ≥ 2.

Developed a refined neck-analysis for energy identity along approximations.

Solved general min-max problems for n-energy with bubbling considerations.

Abstract

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new null-homotopic such maps. The proof follows a similar perturbative argument, which in high dimensions leads to a degenerate and double-phase-type Euler-Lagrange system, making the uniform regularity needed to formalize the bubbling harder to achieve. Then, we develop a refined neck-analysis leading to an energy identity along the approximation, assuming a suitable Struwe-type entropy bound along a sequence of critical points. Finally, we combine these results to solve quite general min-max problems for the $n$-energy modulo bubbling.